- Email: [email protected]
D-spaces, aD-spaces and finite unions
Topology and its Applications 156 (2009) 1459–1462
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
D-spaces, aD-spaces and finite unions Yu Zuoming, Yun Ziqiu ∗,1 Department of Mathematics, Suzhou University, 215006 Suzhou, PR China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 17 December 2007 Received in revised form 29 April 2008 Accepted 26 November 2008 MSC: 54E35 54E99 Keywords: D-space aD-space δθ -refinable Strong Σ -space
In this paper, we prove that if a space X is the union of a finite family of strong Σ -spaces, then X is a D-space. This gives a positive answer to a question posed by Arhangel’skii in [A.V. Arhangel’skii, D-spaces and finite unions, Proc. Amer. Math. Soc. 132 (2004) 2163–2170]. We also obtain results on aD-spaces and finite unions. These results improve the correspond results in [A.V. Arhangel’skii, R.Z. Buzyakova, Addition theorems and D-spaces, Comment. Math. Univ. Carolin. 43 (2002) 653–663] and [Liang-Xue Peng, The D-property of some Lindelöf spaces and related conclusions, Topology Appl. 154 (2007) 469–475]. © 2009 Elsevier B.V. All rights reserved.
1. Introduction
The notion of D-space was introduced by E. van Douwen (see [7]). A neighborhood assignment on a topological space X is a mapping φ of X into the topology τ such that x ∈ φ(x) for each x ∈ X . A space X is called a D-space if for every neighborhood assignment φ on X , there exists a closed discrete [7] (locally finite [3]) subset A of X such that the family φ( A ) = {φ(a): a ∈ A } covers X . A space X is called an aD-space [3] if for each closed subset F of X and each open covering γ of X , there exists a subset A of F , locally finite in F , and a mapping φ of A into γ such that a ∈ φ(a), for each a ∈ A, and φ( A ) covers F . The mapping φ is also called a pointer. Certain generalized metric spaces are known to be D-spaces. For examples, spaces with a point countable weak base or σ -cushioned network (and hence, semi-stratifiable spaces or strong Σ -spaces) are D-spaces [5,9]. We refer the reader to [4,6,8] for more results. Also, spaces with certain covering properties are known to be aD-spaces. For example, T 1 δθ -refinable spaces are aD-spaces [2]. Hence, T 1 θ -refinable spaces and T 1 metaLindelöf spaces are aD-spaces. It was proved by Arhangel’skii that if a space X is the union of a finite family of subparacompact spaces, then X is an aD-space [1]. This result was improved by L. Peng in [11] by showing that if a space X is the union of a finite family of θ -refinable spaces, then X is an aD-space. However, there are few known results on the finite union of what kind of generalized metric spaces is a D-space, though it is known that if a space X is the union of a finite family of metrizable subspaces, then X is a D-space. Arhangel’skii posed the following problem in [1]:
* 1
Corresponding author. E-mail address: [email protected] (Z. Yun). Supported by the National Science Foundation of China (Project No. 10571151).
0166-8641/$ – see front matter doi:10.1016/j.topol.2008.11.017
©
2009 Elsevier B.V. All rights reserved.
1460
Z. Yu, Z. Yun / Topology and its Applications 156 (2009) 1459–1462
Problem 1. ([1, Problem 1.6]) If a space X is the union of a finite collection { X i : i = 1, . . . , k} of Moore spaces, is then X a D-space? In this paper, we answer the above problem positively by proving that if a space X is the union of a finite collection
{ X i : i = 1, . . . , k} of strong Σ -spaces, then X is a D-space. Arhangel’skii also asked in [1] whether the union of two D-spaces (aD-spaces) is a D-space (an aD-space). The following known result can be regarded as a partial answer of this question: If a regular space X = Y ∪ Z , where Y is a paracompact subspace of X and Z is an aD-space, then X is an aD-space [3, Theorem 12]. In this paper, we improve this result by showing that if a T 1 space X = Y ∪ Z , where Y is a metaLindelöf subspace of X and Z is an aD-space, then X is an aD-space. Also, we prove that if a space X is the union of a finite collection { X i : i = 1, . . . , k} of δθ -refinable spaces, then X is an aD-space. This result improves the above-mentioned result of L. Peng. 2. Finite unions and D-spaces Recall that a space is a strong Σ -space if there exists a σ -locally finite family γ of closed sets in X and a cover K of X by compact subsets, such that for any open set U containing an element K of K, K ⊆ Γ ⊆ U for some Γ ∈ γ . Theorem 2.1. Suppose that a space X is the union of a finite collection { X i : i = 1, . . . , k} of strong Σ -spaces, then X is a D-space. Proof. We argue by induction. If k = 1, the statement is obviously true. Assume now that the statement holds for k = n, for some n ∈ ω , and let us show that it is true for k = n + 1. Let φ beany neighborhood assignment of X . For each i = 1, . . . , n + 1, let Ki be a cover of X i by compact subsets of X i , Pi = j <ω Pi j , where Pi is a σ -locally finite collection of closed sets of X i with respect to the definition of strong Σ -spaces, Pi j is locally finite in X i and Pi j ⊆ Pi ( j+1) for each j ∈ ω . For each P ∈ Pi j , i = 1, . . . , n + 1, j ∈ ω , let B P be the collection of all finite subsets F ⊆ X i for which there is a subset, U , of X i , open in X i , such that F ⊆ P \ U ⊆ φ( F ). For each i = 1, . . . , n + 1, put A i1 = {x ∈ X: Pi1 is not locally finite at X }. We can see that A i1 is a closed subset of X , and A i1 ∩ X i = ∅. It follows by the inductive assumption that A i1 is a D-space, so there is a locally finite subset D i1 of A i1 , hence, a locally finitesubset of X , such that A i1 ⊆ φ( D i1 ). For each P ∈ Pi1 , if there exists some B P ∈ B P such that B P ⊆ P \ φ( D i1 ) ⊆ φ( B P ), choose just one B P for P and denote it by B P , else we put B P = ∅. Put Bi1 = { B P : P ∈ Pi1 },
n+1
D 1 = i =1 ( D i1 ∪ Bi1 ), then it is easy to see that D 1 is a locally finite subset of X . Put G 1 = φ( D 1 ). Assume that we have already defined an open subset G j and a locally finite subset D j of X , for some j ∈ ω . For each i = 1, . . . , n + 1, put Pi( j +1) = { P \ G j : P ∈ Pi ( j +1) }, A i ( j +1) = {x ∈ X: Pi( j +1) is not locally finite at X }. We can see that A i ( j +1) is a closed subset of X , and A i ( j +1) ∩ X i = ∅. It follows by the inductive assumption that A i ( j +1) is a D-space since strong Σ -spaces areclosed hereditary. So there is a locally finite subset D i ( j +1) of A i ( j +1) , hence, a locally finite subset of X , such that A ⊆ φ( D ) . For each P ∈ P , if there exists some B ∈ B such that B ⊆ P \ φ( D i ( j +1) ) ∪ G j ⊆ P P P i ( j + 1 ) i ( j + 1 ) i ( j + 1 ) φ( B P ), choose just one set B P for P and denote it by B P , else we put B P = ∅. Put Bi ( j+1) = { B P : P ∈ Pi ( j+1) }, D j +1 =
( D i ( j+1) ∪ Bi ( j +1) ) ∪ D j , then it is easy to see that D j +1 is a locally finite subset of X . Put G j +1 = φ( D j +1 ) ∪ G j . Put D = j <ω D j .
n+1 i =1
Claim 1. X =
d∈ D
φ(d).
Assume the contrary. Then there is an i ∈ {1, . . . , n + 1}, and a K ∈ Ki such that K = K \ d∈ D φ(d) = ∅. Since K is compact in X i , we can find {x1 , x2 , . . . , xk } ⊆ K such that K ⊆ φ({x1 , x2 , . . . , xk }). Put K = K \ φ({x1 , x2 , . . . , xk }). We can find the smallest l such that K ⊆ d∈ D l φ(d). We take the first Pi j containing a P such that K ⊆ P ⊆
φ {x1 , x2 , . . . , xk } ∪ φ(d) ∩ X i ⊆ φ(d) . φ {x1 , x2 , . . . , xk } ∪ d∈ D l
d∈ D l
Let m = max{l, j }, then P ∈ Pim , so {x1 , x2 , . . . , xk } ∈ B P , and we can get a nonempty subset B P of X i in B P with respect to P . Hence P must be covered by G m , a contradiction. Claim 2. D is closed and discrete in X .
For each x ∈ X , x ∈ d∈ D l φ(d) for some l ∈ ω by Claim 1. It means that x is separated from D \ D l by the construction of D l . Meanwhile, D k is locally finite in X for each k l, hence, D is locally finite in X . 2
Z. Yu, Z. Yun / Topology and its Applications 156 (2009) 1459–1462
It is well known that a Moore space is a positively.
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σ -space, and hence a strong Σ -space. Therefore, Theorem 2.1 answers Problem 1
3. Finite unions and aD-spaces Theorem 3.1. If T 1 space X = Y ∪ Z , where Y is a metaLindelöf space and Z is an aD-space, then X is an aD-space. To prove the statement, we need the following lemmas: Lemma 3.2. ([3]) Every closed subspace of an aD-space is an aD-space. We say that a subset D of X distinguishes a collection U of subsets of X if every element of U contains one element of D and every element of D is in exactly one element of U . Lemma 3.3. ([10]) Let U be an open cover of a T 1 space X , let C ⊂ {x ∈ X: ord(x, U ) ω}. Then there is an open partial refinement W of U covering C and a subset D of C such that D distinguishes W and is discrete in X . Proof of Theorem 3.1. Let U be an open cover of X , and let F be a closed subset of X . In view of Lemma 3.2, we can assume that F = X . Since Y is a metaLindelöf space, there is a family V of open subsets of X satisfying: (1) V is point countable at each y ∈ Y ; (2) V refines U ; (3) Y is covered by V .
Let H = X \ V . Clearly, H is closed in X , and H ∩ Y = ∅. Lemma 3.2 implies that H is an aD-space, so we can find a closed discrete inH subset A of H and a pointer f : A → U such that f ( A ) covers H . X\ f ( A ) isclosed in X and is covered by V . Lemma 3.3 implies that Obviously, there exists a closed discrete in g X\ f ( A ) subset B of X \ f ( A ) and a pointer g : B→ U such that ( B ) covers ( X \ f ( A )) ∩ Y . X Z ( A ) ∪ g ( B ) , so \ ( f ( A ) ∪ g ( B )) is contained in and in X . X \ ( f ( A ) ∪ Y f Notice that is covered by closed g ( B )) is also in X \ ( f ( A ) ∪ g ( B )) subset C of an aD-space by Lemma 3.2, so we can find a closed discrete X \ ( f ( A ) ∪ g ( B )) and a pointer h : C → U such that h ( C ) covers X \ ( f ( A ) ∪ g ( B )) . f ( A ) and X \ ( f ( A ) ∪ g ( B )) are closed in X ; A, B and C are disjoint. Now for each x ∈ A ∪ B ∪ C , Clearly, H , X \ let ψ(x) = f (x), if x ∈ A; ψ(x) = g (x), if x ∈ B; and ψ(x) = h(x), if x ∈ C . Obviously, A ∪ B ∪ C is a closed discrete subset of X , and X is covered by ψ( A ∪ B ∪ C ). Thus, X is an aD-space. 2 Corollary 3.4. Suppose that a T 1 space X is the union of a finite collection { X i : i = 1, . . . , k} of metaLindelöf spaces, then X is an aD-space. Proof. We argue by induction. If k = 1, the statement is obviously true. Assume now that the statement holds for some n ∈ ω , and let us show that it is true for k = n + 1. Set Y = X 1 , Z = X 2 ∪ · · · ∪ X n+1 . Then X = Y ∪ Z , and by the assumption of induction, Z is an aD-space. It follows from Theorem 3.1 that X is also an aD-space. 2 In the following, we prove a result which is even more general than Corollary 3.4. Recall that a space X is called δθ -refinable (θ -refinable), if for every open cover U of X , there is an open refinement V = {Vn : n ∈ N } such that: (1) for any n ∈ ω , V is a cover of X ; (2) for any x ∈ X there is some n ∈ N such that ord(x, Vn ) ω (< ω), where ord(x, Vn ) = |{ V ∈ Vn : x ∈ V }|. Theorem 3.5. Suppose that a T 1 space X is the union of a finite collection { X i : i = 1, . . . , k} of δθ -refinable spaces, then X is an aD-space. Proof. We argue by induction. If k = 1, the statement is obviously true. Assume now that the statement holds for k = n, for some n ∈ ω , and let us show that it is true for k = n + 1. Let U be an open cover of X . For each i = 1, . . . , n + 1, let Vi = j <ω Vi j be a collection of open subsets of X satisfying the following conditions: (1) Vi refines U ; (2) X i is covered by Vi j for each j < ω ; (3) for each x ∈ X i , there is some j 0 ∈ ω such that ord(x, Vi j 0 ) ω .
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For each i = 1, . . . , n + 1 and l < ω , let K il = {x ∈ X i : ord(x, Vil ) ω}, F il = X \ Vil . Then X i = l<ω K il , F il is a closed subset of X and F il ∩ X i = ∅. It follows by the inductive assumption that F i1 is an aD-space. Hence, there exists a closed discrete subset D i1 of F i1 , f i1 ( D i1 ), obviously, and a pointer f i1 : D i1 → U , such that F i1 is covered by the subfamily f i1 ( D i1 ) of U . Let H i1 = X \ H i1 is a closed subset of X and covered by Vi1 . By Lemma 3.3, there is a closed discrete subset J i1 of X \ H i1 , and a pointer g i1 : J i1 → U , such that K i1 ∩ ( X \ H i1 ) is covered by the subfamily g i1 ( J i1 ) of U . Notice that both F i1 and X \ H i1 subsets of X , so D i1 and J i1 are closed discrete subsets of X . It is easy to see that D i1 ∩ J i1 = ∅ and are closed K i1 ⊆ f i1 ( D i1 ) ∪ g i1 ( J i1 ) for each i = 1, . . . , n + 1. Note that in the above procedures we can define i1 and J i1 the sets D as well as pointers f i1 andg i1 one after another in such a way that D j1 and J j1 lie outside of f i1 ( D i1 ) ∪ g i1 ( J i1 ) whenever i < j. Put D 1 = { D i1 ∪ J i1 : i = 1, . . . , n + 1}, and define a pointer ψ1 : D 1 → Uby the requirement that the restrictions of ψ1 to D i1 and J i1 coincide with f i1 and g i1 for each i = 1, . . . , n + 1. Put G 1 = ψ1 ( D 1 ), then G 1 is an open subset of X . Let us assume that we have already defined an open subset G l of X , for some l ∈ ω . Repeating the step above, we can get two disjoint closed discrete subsets 1) of X \ G l and two pointers f i (l+1) : D i (l+1) → U and D i (l+1) and J i (l+ g i (l+1) : J i (l+1) → U , such that K i (l+1) \ G l ⊆ f i (l+1) ( D i (l+1) ) ∪ g i (l+1) ( J i (l+1) ). Also, we can define the sets D i (l+1) J f g and as well as pointers and one after another i ( l + 1 ) i ( l + 1 ) i ( l + 1 ) in such a way that D j (l+1) and J j (l+1) lie outside of f i (l+1) ( D i (l+1) ) ∪ g i (l+1) ( J i (l+1) ) whenever i < j. Put D l+1 = { D i (l+1) ∪ J i (l+1) : i = 1, . . . , n + 1}, and define a pointer ψl+1 : D l+1 → U by the requirement that the restrictions of ψl+1 to D i (l+1) and J i (l+1) coincide with f i (l+1) and g i (l+1) for i = 1, . . . , n + 1. Put G l+1 = ψl+1 ( D l+1 ) ∪ G l , then G l+1 is an open subset of X . of K il , for each i = 1, . . . , n + 1 and l ∈ ω , are covered by it. Since The family {G l : l ∈ ω} covers X , since all elements D l ⊆ G l ⊆ G l+1 and D l+1 ∩ G l = ∅, the set D = { D l : l ∈ ω} is closed discrete in X . Thus, the pointer ψ : D → U defined by the requirement that the restriction of ψ to D l coincides with ψl for each l < ω is a pointer we need. 2 By Theorem 3.1, Corollary 3.4 and Theorem 3.5, the following problem raises naturally: Problem 3.6. If a T 1 space X = Y ∪ Z , where Y is a δθ -refinable space and Z is an aD-space, then is X an aD-space? We even do not know the answer of the following problem: Problem 3.7. If a T 1 space X = Y ∪ Z , where Y is a θ -refinable space and Z is an aD-space, then is X an aD-space? Acknowledgement The authors are grateful to the referee for helping us to improve the writing of this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
A.V. Arhangel’skii, D-spaces and finite unions, Proc. Amer. Math. Soc. 132 (2004) 2163–2170. A.V. Arhangel’skii, D-spaces and covering properties, Topology Appl. 146-147 (2005) 437–449. A.V. Arhangel’skii, R.Z. Buzyakova, Addition theorems and D-spaces, Comment. Math. Univ. Carolin. 43 (2002) 653–663. C.R. Borges, A.C. Wehrly, A study of D-spaces, Topology Proc. 16 (1991) 7–15. Dennis K. Burke, Weak-bases and D-spaces, Comment. Math. Univ. Carolin. 48 (2007) 281–289. R.Z. Buzyakova, On D-property of strong Σ spaces, Comment. Math. Univ. Carolin. 43 (2002) 493–495. Erick K. van Douwen, Washek F. Pfeffer, Some properties of Sorgenfrey line and related spaces, Pacific J. Math. 81 (1979) 371–377. Gary Gruenhage, A note on D-spaces, Topology Appl. 153 (2006) 2229–2240. Shou Lin, A note on D-spaces, Comment. Math. Univ. Carolin. 42 (2006) 313–316. J.D. Mashburn, A note on irreducibility and weak covering properties, Topology Proc. 9 (1984) 339–352. Liang-Xue Peng, The D-property of some Lindelöf spaces and related conclusions, Topology Appl. 154 (2007) 469–475.
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
D-spaces, aD-spaces and finite unions Yu Zuoming, Yun Ziqiu ∗,1 Department of Mathematics, Suzhou University, 215006 Suzhou, PR China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 17 December 2007 Received in revised form 29 April 2008 Accepted 26 November 2008 MSC: 54E35 54E99 Keywords: D-space aD-space δθ -refinable Strong Σ -space
In this paper, we prove that if a space X is the union of a finite family of strong Σ -spaces, then X is a D-space. This gives a positive answer to a question posed by Arhangel’skii in [A.V. Arhangel’skii, D-spaces and finite unions, Proc. Amer. Math. Soc. 132 (2004) 2163–2170]. We also obtain results on aD-spaces and finite unions. These results improve the correspond results in [A.V. Arhangel’skii, R.Z. Buzyakova, Addition theorems and D-spaces, Comment. Math. Univ. Carolin. 43 (2002) 653–663] and [Liang-Xue Peng, The D-property of some Lindelöf spaces and related conclusions, Topology Appl. 154 (2007) 469–475]. © 2009 Elsevier B.V. All rights reserved.
1. Introduction
The notion of D-space was introduced by E. van Douwen (see [7]). A neighborhood assignment on a topological space X is a mapping φ of X into the topology τ such that x ∈ φ(x) for each x ∈ X . A space X is called a D-space if for every neighborhood assignment φ on X , there exists a closed discrete [7] (locally finite [3]) subset A of X such that the family φ( A ) = {φ(a): a ∈ A } covers X . A space X is called an aD-space [3] if for each closed subset F of X and each open covering γ of X , there exists a subset A of F , locally finite in F , and a mapping φ of A into γ such that a ∈ φ(a), for each a ∈ A, and φ( A ) covers F . The mapping φ is also called a pointer. Certain generalized metric spaces are known to be D-spaces. For examples, spaces with a point countable weak base or σ -cushioned network (and hence, semi-stratifiable spaces or strong Σ -spaces) are D-spaces [5,9]. We refer the reader to [4,6,8] for more results. Also, spaces with certain covering properties are known to be aD-spaces. For example, T 1 δθ -refinable spaces are aD-spaces [2]. Hence, T 1 θ -refinable spaces and T 1 metaLindelöf spaces are aD-spaces. It was proved by Arhangel’skii that if a space X is the union of a finite family of subparacompact spaces, then X is an aD-space [1]. This result was improved by L. Peng in [11] by showing that if a space X is the union of a finite family of θ -refinable spaces, then X is an aD-space. However, there are few known results on the finite union of what kind of generalized metric spaces is a D-space, though it is known that if a space X is the union of a finite family of metrizable subspaces, then X is a D-space. Arhangel’skii posed the following problem in [1]:
* 1
Corresponding author. E-mail address: [email protected] (Z. Yun). Supported by the National Science Foundation of China (Project No. 10571151).
0166-8641/$ – see front matter doi:10.1016/j.topol.2008.11.017
©
2009 Elsevier B.V. All rights reserved.
1460
Z. Yu, Z. Yun / Topology and its Applications 156 (2009) 1459–1462
Problem 1. ([1, Problem 1.6]) If a space X is the union of a finite collection { X i : i = 1, . . . , k} of Moore spaces, is then X a D-space? In this paper, we answer the above problem positively by proving that if a space X is the union of a finite collection
{ X i : i = 1, . . . , k} of strong Σ -spaces, then X is a D-space. Arhangel’skii also asked in [1] whether the union of two D-spaces (aD-spaces) is a D-space (an aD-space). The following known result can be regarded as a partial answer of this question: If a regular space X = Y ∪ Z , where Y is a paracompact subspace of X and Z is an aD-space, then X is an aD-space [3, Theorem 12]. In this paper, we improve this result by showing that if a T 1 space X = Y ∪ Z , where Y is a metaLindelöf subspace of X and Z is an aD-space, then X is an aD-space. Also, we prove that if a space X is the union of a finite collection { X i : i = 1, . . . , k} of δθ -refinable spaces, then X is an aD-space. This result improves the above-mentioned result of L. Peng. 2. Finite unions and D-spaces Recall that a space is a strong Σ -space if there exists a σ -locally finite family γ of closed sets in X and a cover K of X by compact subsets, such that for any open set U containing an element K of K, K ⊆ Γ ⊆ U for some Γ ∈ γ . Theorem 2.1. Suppose that a space X is the union of a finite collection { X i : i = 1, . . . , k} of strong Σ -spaces, then X is a D-space. Proof. We argue by induction. If k = 1, the statement is obviously true. Assume now that the statement holds for k = n, for some n ∈ ω , and let us show that it is true for k = n + 1. Let φ beany neighborhood assignment of X . For each i = 1, . . . , n + 1, let Ki be a cover of X i by compact subsets of X i , Pi = j <ω Pi j , where Pi is a σ -locally finite collection of closed sets of X i with respect to the definition of strong Σ -spaces, Pi j is locally finite in X i and Pi j ⊆ Pi ( j+1) for each j ∈ ω . For each P ∈ Pi j , i = 1, . . . , n + 1, j ∈ ω , let B P be the collection of all finite subsets F ⊆ X i for which there is a subset, U , of X i , open in X i , such that F ⊆ P \ U ⊆ φ( F ). For each i = 1, . . . , n + 1, put A i1 = {x ∈ X: Pi1 is not locally finite at X }. We can see that A i1 is a closed subset of X , and A i1 ∩ X i = ∅. It follows by the inductive assumption that A i1 is a D-space, so there is a locally finite subset D i1 of A i1 , hence, a locally finitesubset of X , such that A i1 ⊆ φ( D i1 ). For each P ∈ Pi1 , if there exists some B P ∈ B P such that B P ⊆ P \ φ( D i1 ) ⊆ φ( B P ), choose just one B P for P and denote it by B P , else we put B P = ∅. Put Bi1 = { B P : P ∈ Pi1 },
n+1
D 1 = i =1 ( D i1 ∪ Bi1 ), then it is easy to see that D 1 is a locally finite subset of X . Put G 1 = φ( D 1 ). Assume that we have already defined an open subset G j and a locally finite subset D j of X , for some j ∈ ω . For each i = 1, . . . , n + 1, put Pi( j +1) = { P \ G j : P ∈ Pi ( j +1) }, A i ( j +1) = {x ∈ X: Pi( j +1) is not locally finite at X }. We can see that A i ( j +1) is a closed subset of X , and A i ( j +1) ∩ X i = ∅. It follows by the inductive assumption that A i ( j +1) is a D-space since strong Σ -spaces areclosed hereditary. So there is a locally finite subset D i ( j +1) of A i ( j +1) , hence, a locally finite subset of X , such that A ⊆ φ( D ) . For each P ∈ P , if there exists some B ∈ B such that B ⊆ P \ φ( D i ( j +1) ) ∪ G j ⊆ P P P i ( j + 1 ) i ( j + 1 ) i ( j + 1 ) φ( B P ), choose just one set B P for P and denote it by B P , else we put B P = ∅. Put Bi ( j+1) = { B P : P ∈ Pi ( j+1) }, D j +1 =
( D i ( j+1) ∪ Bi ( j +1) ) ∪ D j , then it is easy to see that D j +1 is a locally finite subset of X . Put G j +1 = φ( D j +1 ) ∪ G j . Put D = j <ω D j .
n+1 i =1
Claim 1. X =
d∈ D
φ(d).
Assume the contrary. Then there is an i ∈ {1, . . . , n + 1}, and a K ∈ Ki such that K = K \ d∈ D φ(d) = ∅. Since K is compact in X i , we can find {x1 , x2 , . . . , xk } ⊆ K such that K ⊆ φ({x1 , x2 , . . . , xk }). Put K = K \ φ({x1 , x2 , . . . , xk }). We can find the smallest l such that K ⊆ d∈ D l φ(d). We take the first Pi j containing a P such that K ⊆ P ⊆
φ {x1 , x2 , . . . , xk } ∪ φ(d) ∩ X i ⊆ φ(d) . φ {x1 , x2 , . . . , xk } ∪ d∈ D l
d∈ D l
Let m = max{l, j }, then P ∈ Pim , so {x1 , x2 , . . . , xk } ∈ B P , and we can get a nonempty subset B P of X i in B P with respect to P . Hence P must be covered by G m , a contradiction. Claim 2. D is closed and discrete in X .
For each x ∈ X , x ∈ d∈ D l φ(d) for some l ∈ ω by Claim 1. It means that x is separated from D \ D l by the construction of D l . Meanwhile, D k is locally finite in X for each k l, hence, D is locally finite in X . 2
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It is well known that a Moore space is a positively.
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σ -space, and hence a strong Σ -space. Therefore, Theorem 2.1 answers Problem 1
3. Finite unions and aD-spaces Theorem 3.1. If T 1 space X = Y ∪ Z , where Y is a metaLindelöf space and Z is an aD-space, then X is an aD-space. To prove the statement, we need the following lemmas: Lemma 3.2. ([3]) Every closed subspace of an aD-space is an aD-space. We say that a subset D of X distinguishes a collection U of subsets of X if every element of U contains one element of D and every element of D is in exactly one element of U . Lemma 3.3. ([10]) Let U be an open cover of a T 1 space X , let C ⊂ {x ∈ X: ord(x, U ) ω}. Then there is an open partial refinement W of U covering C and a subset D of C such that D distinguishes W and is discrete in X . Proof of Theorem 3.1. Let U be an open cover of X , and let F be a closed subset of X . In view of Lemma 3.2, we can assume that F = X . Since Y is a metaLindelöf space, there is a family V of open subsets of X satisfying: (1) V is point countable at each y ∈ Y ; (2) V refines U ; (3) Y is covered by V .
Let H = X \ V . Clearly, H is closed in X , and H ∩ Y = ∅. Lemma 3.2 implies that H is an aD-space, so we can find a closed discrete inH subset A of H and a pointer f : A → U such that f ( A ) covers H . X\ f ( A ) isclosed in X and is covered by V . Lemma 3.3 implies that Obviously, there exists a closed discrete in g X\ f ( A ) subset B of X \ f ( A ) and a pointer g : B→ U such that ( B ) covers ( X \ f ( A )) ∩ Y . X Z ( A ) ∪ g ( B ) , so \ ( f ( A ) ∪ g ( B )) is contained in and in X . X \ ( f ( A ) ∪ Y f Notice that is covered by closed g ( B )) is also in X \ ( f ( A ) ∪ g ( B )) subset C of an aD-space by Lemma 3.2, so we can find a closed discrete X \ ( f ( A ) ∪ g ( B )) and a pointer h : C → U such that h ( C ) covers X \ ( f ( A ) ∪ g ( B )) . f ( A ) and X \ ( f ( A ) ∪ g ( B )) are closed in X ; A, B and C are disjoint. Now for each x ∈ A ∪ B ∪ C , Clearly, H , X \ let ψ(x) = f (x), if x ∈ A; ψ(x) = g (x), if x ∈ B; and ψ(x) = h(x), if x ∈ C . Obviously, A ∪ B ∪ C is a closed discrete subset of X , and X is covered by ψ( A ∪ B ∪ C ). Thus, X is an aD-space. 2 Corollary 3.4. Suppose that a T 1 space X is the union of a finite collection { X i : i = 1, . . . , k} of metaLindelöf spaces, then X is an aD-space. Proof. We argue by induction. If k = 1, the statement is obviously true. Assume now that the statement holds for some n ∈ ω , and let us show that it is true for k = n + 1. Set Y = X 1 , Z = X 2 ∪ · · · ∪ X n+1 . Then X = Y ∪ Z , and by the assumption of induction, Z is an aD-space. It follows from Theorem 3.1 that X is also an aD-space. 2 In the following, we prove a result which is even more general than Corollary 3.4. Recall that a space X is called δθ -refinable (θ -refinable), if for every open cover U of X , there is an open refinement V = {Vn : n ∈ N } such that: (1) for any n ∈ ω , V is a cover of X ; (2) for any x ∈ X there is some n ∈ N such that ord(x, Vn ) ω (< ω), where ord(x, Vn ) = |{ V ∈ Vn : x ∈ V }|. Theorem 3.5. Suppose that a T 1 space X is the union of a finite collection { X i : i = 1, . . . , k} of δθ -refinable spaces, then X is an aD-space. Proof. We argue by induction. If k = 1, the statement is obviously true. Assume now that the statement holds for k = n, for some n ∈ ω , and let us show that it is true for k = n + 1. Let U be an open cover of X . For each i = 1, . . . , n + 1, let Vi = j <ω Vi j be a collection of open subsets of X satisfying the following conditions: (1) Vi refines U ; (2) X i is covered by Vi j for each j < ω ; (3) for each x ∈ X i , there is some j 0 ∈ ω such that ord(x, Vi j 0 ) ω .
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For each i = 1, . . . , n + 1 and l < ω , let K il = {x ∈ X i : ord(x, Vil ) ω}, F il = X \ Vil . Then X i = l<ω K il , F il is a closed subset of X and F il ∩ X i = ∅. It follows by the inductive assumption that F i1 is an aD-space. Hence, there exists a closed discrete subset D i1 of F i1 , f i1 ( D i1 ), obviously, and a pointer f i1 : D i1 → U , such that F i1 is covered by the subfamily f i1 ( D i1 ) of U . Let H i1 = X \ H i1 is a closed subset of X and covered by Vi1 . By Lemma 3.3, there is a closed discrete subset J i1 of X \ H i1 , and a pointer g i1 : J i1 → U , such that K i1 ∩ ( X \ H i1 ) is covered by the subfamily g i1 ( J i1 ) of U . Notice that both F i1 and X \ H i1 subsets of X , so D i1 and J i1 are closed discrete subsets of X . It is easy to see that D i1 ∩ J i1 = ∅ and are closed K i1 ⊆ f i1 ( D i1 ) ∪ g i1 ( J i1 ) for each i = 1, . . . , n + 1. Note that in the above procedures we can define i1 and J i1 the sets D as well as pointers f i1 andg i1 one after another in such a way that D j1 and J j1 lie outside of f i1 ( D i1 ) ∪ g i1 ( J i1 ) whenever i < j. Put D 1 = { D i1 ∪ J i1 : i = 1, . . . , n + 1}, and define a pointer ψ1 : D 1 → Uby the requirement that the restrictions of ψ1 to D i1 and J i1 coincide with f i1 and g i1 for each i = 1, . . . , n + 1. Put G 1 = ψ1 ( D 1 ), then G 1 is an open subset of X . Let us assume that we have already defined an open subset G l of X , for some l ∈ ω . Repeating the step above, we can get two disjoint closed discrete subsets 1) of X \ G l and two pointers f i (l+1) : D i (l+1) → U and D i (l+1) and J i (l+ g i (l+1) : J i (l+1) → U , such that K i (l+1) \ G l ⊆ f i (l+1) ( D i (l+1) ) ∪ g i (l+1) ( J i (l+1) ). Also, we can define the sets D i (l+1) J f g and as well as pointers and one after another i ( l + 1 ) i ( l + 1 ) i ( l + 1 ) in such a way that D j (l+1) and J j (l+1) lie outside of f i (l+1) ( D i (l+1) ) ∪ g i (l+1) ( J i (l+1) ) whenever i < j. Put D l+1 = { D i (l+1) ∪ J i (l+1) : i = 1, . . . , n + 1}, and define a pointer ψl+1 : D l+1 → U by the requirement that the restrictions of ψl+1 to D i (l+1) and J i (l+1) coincide with f i (l+1) and g i (l+1) for i = 1, . . . , n + 1. Put G l+1 = ψl+1 ( D l+1 ) ∪ G l , then G l+1 is an open subset of X . of K il , for each i = 1, . . . , n + 1 and l ∈ ω , are covered by it. Since The family {G l : l ∈ ω} covers X , since all elements D l ⊆ G l ⊆ G l+1 and D l+1 ∩ G l = ∅, the set D = { D l : l ∈ ω} is closed discrete in X . Thus, the pointer ψ : D → U defined by the requirement that the restriction of ψ to D l coincides with ψl for each l < ω is a pointer we need. 2 By Theorem 3.1, Corollary 3.4 and Theorem 3.5, the following problem raises naturally: Problem 3.6. If a T 1 space X = Y ∪ Z , where Y is a δθ -refinable space and Z is an aD-space, then is X an aD-space? We even do not know the answer of the following problem: Problem 3.7. If a T 1 space X = Y ∪ Z , where Y is a θ -refinable space and Z is an aD-space, then is X an aD-space? Acknowledgement The authors are grateful to the referee for helping us to improve the writing of this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
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